Some people have a very difficult time stepping outside themselves to do what is right if that thing causes them pain. I'm assuming he'll grieve the dog being gone and this causes him pain and so he isn't going to do it - even though he knows it is the right thing and will happen within days anyway. He will increase the dogs suffering to trya and deny and avoid his own.
Just do the right thing yourself.
Years ago I was in a very very similar situation. The poor dog was suffering and my ex would not take her in. He just couldn't/wouldn't do it. He ran from that responsibility, avoided it, and let the dog suffer. It wasn't my dog, we weren't even married but we were living together. I took her to the vet and put her down. He wasn't even mad at me. He was relieved and the dog was spared another month of misery.
I haven't thought of that in years. It was a very telling moment. He was showing me who he was and I was not paying attention. That basic method of dealing with things has repeated itself over and over and the stakes are much higher now.